The technique of Kolyvagin used to control the size of arithmetic Selmer groups has, as input, a certain system of cohomological classes (or rational points, or algebraic cycles, or elements in algebraic K-theory, depending upon the setting) over abelian extensions of the base field. Such systems have been called "Euler Systems" by Kolyvagin. As an intermediary step in Kolyvagin's method, a given Euler System is used to construct a specific collection of cohomology classes over the base field. It is this collection of cohomology classes over the base field that directly provide sufficiently many "local linear relations" satisfied by elements in the Selmer group so as to obtain good upper bounds for the size of the Selmer group. Such "collections of cohomology classes over the base field" enjoy (somewhat surprisingly) very rigid inter-relations. Karl Rubin and I have a manuscript in which we study this type of structure ; we call collections of cohomology classes satisfying these relations "Kolyvagin Systems". These Kolyvagin Systems turn out to have especially nice, and interesting, properties. For one thing, they seem more congenial to Galois deformations than are Euler systems, and they can be constructed in instances where we have no construction, yet, of a corresponding Euler system. It is helpful to view the generator of the module of Kolyvagin Systems, when there is only one generator, as something of a p-adic L-function (with values in cohomology groups) governing the arithmetic of the situtation. The aim of these two lectures is to elucidate this structure and its basic arithmetic applications.

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